Abstract

I will present some recent results concerning the higher gradient inte- grability of σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of div(σ∇u) = 0. When σ is assumed to be symmetric, then the optimal integrability exponent of the gradient field is known thanks to the work of Astala and Leonetti & Nesi. I will discuss the case when only the ellipticity is fixed and σ is otherwise unconstrained and show that the optimal exponent is attained on the class of two-phase conductivities σ:Ω⊂R2 →(σ1,σ2)⊂M2×2. The optimal exponent is established, in the strongest possible way of the existence of so-called exact solutions, via the exhibition of optimal microgeometries. (Joint work with V. Nesi and M. Ponsiglione.)